MAT 21C – Lecture 7 – Formulas and Variations in Series PDF

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Professor: David Cherney...

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MAT 21C – Lecture 7 – Formulas and Variations in Series 

Important Identities to Remember ∞ 1 o a) converges to ∑ x n for x in (-1, 1) 1−x n=0 ∞

o b) e x 

1 n x ∑ n=0 n !

converges to

for x in

(−∞ , ∞)

There are variations of these identities via 1. Multiplication ∞

∞

∞

a. Example:

1 x n +1 converges ¿ x ∑ x n =∑ x∗x n=∑ x =x 1−x 1−x n=0 n=0 n=0 ∞

∑ xn

or by re-indexing such that n starts at 1, then

n=1 ∞

∞

∞

x n+1 x x∗x n . =∑ =∑ n=0 n ! n=0 n ! n=0 n !

x∑

b. Example 2: x e x converges to

n

∞

By re-indexing with n starting at 1, we obtain

xk . ∑ k=1 (n−1)!

2. Differentiation and integration a. Example: ∞

∞

1 1 dx converges ¿ ∫∑ x n dx ⟺−ln ( 1−x ) converges ¿ ∑ x ∫ 1−x n=0 n=0 n+1 ∞

1 x ∑ n=0 n+1

b. By re-indexing,

n+ 1

∞

1 k x . k=1 k

=∑

∞

c. Thus,

−ln ( 1−x ) converges to

∑ n1 x

n

for x in [-1, 1).

n=1

3. Substitution for translation and rescaling 1 1 = , which converges to a. Example: x 1−( 1−x ) ∞

∞

n=0

n=0

∑ (1−x )n=∑(−1)n (x−1)n . ∞

∞

1 n n converges to ∑ (3 x)n=∑ 3 x 1−3 x n=0 n=0 1 1 = c. Example 3: converges to 2 2 1+ x 1−(− x ) b. Example 2:

∞

n=0

∞

∞

∑ (− x 2)n=∑ (−1 )n x 2 n n=0

. Then

2 n +1

x (−1 )n ∫ 1+1x2 dx=∑ n=0 2n+ 1

.

n+1

.

∞

∞

n=0

n=0

and

−3

∞

∑ en ! x n

e−3 converges to

e x−3= e x∗¿

n

∑ n1! (5 x)n=∑ n5! x n

d. Example 4: e 5 x converges to

.

n=0 ∞



n

2

8 n the terms ( ) → 0 fast 10

8 ) +… ∑ (108 ) =1+ 108 +( 10

For the series

n=0

| |

8 n+1 ) 8 10 =...